DepartmentofMathematicsMath112Fall2001FinalExamNAME:_________________________________________Instructor:_______________________Instructions:Thereare11pagesonthisexam.Theyareweightedequally.Choose10pagestobegraded.Crossoutthepageyoudonotwantcounted.Showallworkclearlytoreceivefullcredit.Youmustuseprinciplesofcalculustosupportallanswers.Ifyouneedmorespaceforyoursolutions,usethebackofoneofthepages.Youmustindicateclearlywhereyourextraworkcanbefound.Besuretoneatlycrossoutallworkthatyoudonotwanttobeconsidered.Otherwiseitmayhavetheeffectofreducingyourscore.1.(a)Showwhetherthefollowingimproperintegralconvergesordiverges.Ifitconverges,evaluatetheintegral.;041xxdx(b)Showwhetherthisimproperintegralconvergesordiverges.;1.cos2xx31dx2.Evaluateeachintegral.a);xsin2xdxb);2e"x2dxc);dxx3x23.Foreachofthefollowingmethods,clearlyexplain,withoutdoingnumericalcalculations,whetheritgivesanoverestimateorunderestimateoftheintegral;01ex3dxa)left-pointmethodb)right-pointmethodc)midpointmethod,d)trapezoidrule4.(a).Evaluatetheworkneededtopumpthewateroutofarectangularbasementofthedimensions:depth3m,width10m,length15m.Thebasementisfullofwater,thepumpis1mabovethebasement.(Note:Themassofwateris1000kg/m3andg9.8m/s2.(b)Findthehydrostaticforceononeofthe10mby3mwallsofthebasementfrompart(a)whenthebasementisfullofwater.5.Considertheareaboundedbythefollowingcurvesthataregraphedbelow:x4yy24andyx.Graphsofx4yy24,yx.(a)Findtheareaoftheregionenclosedbythetwocurves.(b)Evaluatethevolumeofthesolidobtainedbyrotationoftheenclosedareaofpart(a)aboutthey-axis.6.(a)Computetheexactlengthofthecurvegivenparametricallybyx13t3,y12t2for0JtJ2.(b)Thevolume,V,inlitersofairinthelungsduringa5-secondrespiratorycycleisapproximatedbythemodelV0.1729t0.1522t20.0374t3,wheretisthetimeinseconds.Setuptheintegralneededtoapproximatetheaveragevolumeofairinthelungsduringone5-secondcycle.(Donotevaluatetheintegral.)7.a)Solvethedifferentialequationdydx2x2xygiventhaty 04.(Yoursolutionshouldbeintheformy___.b)Atankcontains500litersofbrinewith10kgofdissolvedsalt.Purewaterentersthetankatarateof10litersperminute.Thewell-mixedsolutiondrainsfromthetankatarateof8litersperminute.Howmuchsaltisinthetankafter10minutes?8.ConsiderthedifferentialequationdPdt0.06P 1P800asamodelforafishpopulationinLakeWobegon,wheretismeasuredinweeks.Thereare200fishinthelakeattimet0.a)Findthesolutionforthedifferentialequation.b)Findthenumberoffishinthelakeafter21weeks.c)Whenisthepopulationgrowingmostrapidly?d)Whenwilltherebe300fishinthelake?d)SketchthegraphofP t,where0JtJ100.9.(a)Calculatethesumoftheinfiniteseries!n1.2 34n.(b) Show whether the series converges or diverges. State the name of the test used.!n1.12n1(c) Show whether the following series converges or diverges. If it converges, does it convergeabsolutely?!n1. 1nn3410.Considerthefollowingpowerseries:f x!n0. x3n5na)Findtheintervalofconvergenceoff x.b)Findtheradiusofconvergenceoff x.c)FindtheintervalofconvergenceoffU x.11.(a)WritetheMaclaurinseriesforf xe"x.b)FindatleastthefirstthreetermsoftheMaclaurinseriesforf